Problem 24 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-27)^{2 / 3} $$

Step-by-Step Solution

Answer

The expression \((-27)^{2/3}\) evaluates to 9.

Step 1: Understand the Expression

The expression \((-27)^{2/3}\) involves taking the cube root of -27 and then squaring the result. This is because the exponent \(2/3\) can be interpreted as the cube root (denominator) of (-27), which is then raised to the power of 2 (numerator).

Step 2: Evaluate the Cube Root

Find \((-27)^{1/3}\), which is the cube root of -27. Since \((-3)^3 = -27\), the cube root of -27 is -3.

Step 3: Square the Result

Take the result from Step 2, which is -3, and square it to find \((-3)^2\). Squaring -3 gives 9 since \((-3) imes (-3) = 9\).

Step 4: Combine the Results

Therefore, \((-27)^{2/3}\) simplifies to 9.

Key Concepts

Cube Roots

A cube root is when you find a number that multiplies by itself three times to equal a given number. When it comes to negative numbers, you can take the cube root without an issue, unlike even roots. This is because the product of a negative times a negative is a positive, and that positive times another negative yields a negative.

For example:

• The cube root of \(-27\) is \(-3\), since \((-3) \times (-3) \times (-3) = -27\).
• This means that the cube root of a negative number will always be negative.

Cube roots are special because they maintain the sign of the number inside the root. Understanding this helps in evaluating expressions and simplifying them accurately.

Fractional Exponents

Fractional exponents offer a concise way to denote roots and powers simultaneously. In the expression \((-27)^{2/3}\), the exponent \(2/3\) indicates both cube root and squaring. The denominator \(3\) tells us to take the cube root, while the numerator \(2\) indicates to square the result.

This is interpreted as:

• First, take the cube root of \(-27\) to get \(-3\).
• Then, take this result and square it to yield \((-3)^2 = 9\).

This breaks down the process into manageable steps, making complex expressions easier to work with while using fractional exponents.

Negative Numbers

Dealing with negative numbers can seem tricky, especially in the context of exponents and roots. It's essential to remember that multiplying two negative numbers results in a positive number, but multiplying a positive by a negative stays negative.

For instance:

• a negative number squared, like \((-3)^2\) results in \(9\).
• However, when you have a cube root of a negative, such as \((-27)^{1/3}\), the result is negative, delivering \(-3\).

Understanding these properties ensures correct handling of negatives in calculation, especially when they interact with roots and powers.

Exponentiation Steps

Exponentiation involves raising numbers to powers, which can be simplified using step-by-step techniques for better comprehension. This systematic approach is beneficial for complex numbers or those involving roots or fractional powers.

In the expression \((-27)^{2/3}\), the steps are straightforward:

• Begin by finding the cube root of \(-27\), resulting in \(-3\).
• Then, take this result and square it, which gives \(9\).
• Finally, integrate these operations: cube rooting, followed by squaring. Always tackle the root first, as determined by the denominator of the fractional exponent.

These steps guide the simplification process, ensuring clarity and accuracy in obtaining the final result.